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Create regression model with ARIMA time series errors
regARIMA creates a regression model with ARIMA time series errors to maintain the sensitivity interpretation of regression coefficients.
By default, the time series errors (also called unconditional disturbances) are independent, identically distributed, mean 0 Gaussian random variables. If the errors have an autocorrelation structure, then you can specify models for them. The models include:
moving average (MA)
autoregressive (AR)
mixed autoregressive and moving average (ARMA)
integrated (ARIMA)
multiplicative seasonal (SARIMA)
Specify error models containing known coefficients to:
Mdl = regARIMA creates a regression model with degree 0 ARIMA errors and no regression coefficient.
Mdl = regARIMA(p,D,q) creates a regression model with errors modeled by a nonseasonal, linear time series with autoregressive degree p, differencing degree D, and moving average degree q.
Mdl = regARIMA(Name,Value) creates a regression model with ARIMA errors using additional options specified by one or more Name,Value pair arguments. Name can also be a property name and Value is the corresponding value. Name must appear inside single quotes (''). You can specify several Name,Value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Note: For regression models with nonseasonal ARIMA errors, use p, D, and q. For regression models with seasonal ARIMA errors, use Name,Value pair arguments. 
Specify optional commaseparated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.
Notes

AR 
Cell vector of nonseasonal, autoregressive coefficients corresponding to a stable polynomial of the error model. Associated lags are 1,2,...,p, which is the nonseasonal, autoregressive polynomial degree, or as specified in ARLags. 
Beta 
Real vector of regression coefficients corresponding to the columns of the predictor data matrix. 
D 
Nonnegative integer indicating the nonseasonal integration degree of the error model. 
Distribution 
Data structure for the conditional probability distribution of the innovation process. The field Name stores the distribution name 'Gaussian' or 't'. If the distribution is 't', then the structure also has the field DoF that stores the degrees of freedom. 
Intercept 
Scalar intercept in the error model. 
MA 
Cell vector of nonseasonal moving average coefficients corresponding to an invertible polynomial of the error model. Associated lags are 1,2,...,q to the degree of the nonseasonal moving average polynomial, or as specified in MALags. 
P 
Scalar, compound autoregressive polynomial degree of the error model. P is the total number of lagged observations necessary to initialize the autoregressive component of the error model. P includes the effects of nonseasonal and seasonal integration captured by the properties D and Seasonality, respectively, and the nonseasonal and seasonal autoregressive polynomials AR and SAR, respectively. P does not necessarily conform to standard Box and Jenkins notation [1]. If D = 0, Seasonality = 0, and SAR = {}, then P conforms to the standard notation. 
Q 
Scalar, compound moving average polynomial degree of the error model. Q is the total number of lagged innovations necessary to initialize the moving average component of the model. Q includes the effects of nonseasonal and seasonal moving average polynomials MA and SMA, respectively. Q does not necessarily conform to standard Box and Jenkins notation [1]. If SMA = {}, then Q conforms to the standard notation. 
SAR 
Cell vector of seasonal autoregressive coefficients corresponding to a stable polynomial of the error model. Associated lags are 1,2,...,p_{s}, which is the seasonal autoregressive polynomial degree, or as specified in SARLags. 
SMA 
Cell vector of seasonal moving average coefficients corresponding to an invertible polynomial of the error model. Associated lags are 1,2,...,q_{s}, which is the seasonal moving average polynomial degree, or as specified in SMALags. 
Seasonality 
Nonnegative integer indicating the seasonal integration degree of the error model. 
Variance 
Positive scalar variance of the model innovations. 
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impulse  Impulse response of regression model with ARIMA errors 
infer  Infer innovations of regression models with ARIMA errors 
Display estimation results for regression models with ARIMA errors  
simulate  Monte Carlo simulation of regression model with ARIMA errors 
A model that explains the behavior of a response using a linear regression model with predictor data, though the errors have autocorrelation indicative of an ARIMA process.
The model has the following form (in lag operator notation):
$$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a\left(L\right)A\left(L\right){\left(1L\right)}^{D}\left(1{L}^{s}\right){u}_{t}=b\left(L\right)B\left(L\right){\epsilon}_{t},\end{array}$$
where
t = 1,...,T.
y_{t} is the response series.
X_{t} is row t of X, which is the matrix of concatenated predictor data vectors. That is, X_{t} is observation t of each predictor series.
c is the regression model intercept.
β is the regression coefficient.
u_{t} is the disturbance series.
ε_{t} is the innovations series.
$${L}^{j}{y}_{t}={y}_{tj}.$$
$$a\left(L\right)=\left(1{a}_{1}L\mathrm{...}{a}_{p}{L}^{p}\right),$$ which is the degree p, nonseasonal autoregressive polynomial.
$$A\left(L\right)=\left(1{A}_{1}L\mathrm{...}{A}_{{p}_{s}}{L}^{{p}_{s}}\right),$$ which is the degree p_{s}, seasonal autoregressive polynomial.
$${\left(1L\right)}^{D},$$ which is the degree D, nonseasonal integration polynomial.
$$\left(1{L}^{s}\right),$$ which is the degree s, seasonal integration polynomial.
$$b\left(L\right)=\left(1+{b}_{1}L+\mathrm{...}+{b}_{q}{L}^{q}\right),$$ which is the degree q, nonseasonal moving average polynomial.
$$B\left(L\right)=\left(1+{B}_{1}L+\mathrm{...}+{B}_{{q}_{s}}{L}^{{q}_{s}}\right),$$ which is the degree q_{s}, seasonal moving average polynomial.
Regression models with ARIMA errors contain a hierarchy of error series. The unconditional disturbance, u_{t}, or structural disturbance, is based on the structural regression component. The conditional error (onestepahead forecast or prediction error), ε_{t} is the innovation of u_{t}.
Note: The degrees of the lag operators in the seasonal polynomials A(L) and B(L) do not conform to those defined by Box and Jenkins [1]. In other words, Econometrics Toolbox™ does not treat p_{1} = s, p_{2} = 2s,...,p_{s} = c_{p}s nor q_{1} = s, q_{2} = 2s,...,q_{s} = c_{q}s where c_{p} and c_{q} are positive integers. The software is flexible as it lets you specify the lag operator degrees. See Multiplicative ARIMA Model Specifications. 
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.
[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.